Lectures on Differential and Integral EquationsPDF电子书下载

Chapter 1．THE INITIAL VALUE PROBLEM FOR ORDINARY DIFFERENTIAL EQUATIONS 1

1．Successive Approximations 1

1．Existence and uniqueness of the solution of the ordinary differential equation of the first order 1

2．Remark on approximate solutions 6

3．Integration constants 8

4．Solution by power series expansion 10

5．Differential equations containing parameters．Perturbation theory 14

6．Existence and uniqueness of the solution of a system of differential equations 17

2．Linear Differential Equations of the nth Order 21

7．Singular points for linear differential equations 21

8．Fundamental system of solutions 23

9．Wronskian．Liouville＇s formula 27

10．Lagrange＇s method of variation of constants and D＇Alembert＇s method of reduction of order 29

11．Linear differential equations with constant coefficients 31

3．Second Order Differential Equations of the Fuchs Type 37

12．Regular singular points．Fuchs＇theorem 37

13．Gauss differential equations 45

14．Legendre differential equations 48

15．Bessel differential equations 51

Chapter 2．THE BOUNDARY VALUE PROBLEM FOR LINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDER 61

1．Boundary Value Problem 61

16．Boundary value problem of the Sturm-Liouville type 61

17．Green＇s function．Reduction to integral equations 64

18．Periodic solutions．Generalized Green＇s function 68

2．Hilbert-Schmidt Theory of Integral Equations with Symmetric Kernels 76

19．The Ascoli-Arzelà theorem 76

20．Existence proof for the eigenvalues 80

21．The Bessel inequality．The Hilbert-Schmidt expansion theorem 83

22．Approximations of eigenvalues．Rayleigh＇s principle and the Kryloff-Weinstein theorem 91

23．Inhomogeneous integral equations 96

24．Hermite，Laguerre and Legendre polynomials 100

3．Asymptotic Expression of Eigenvalues and Eigenfunctions，Liouville＇s Method 110

25．The Liouville transformation 110

26．Asymptotic expressions of eigenvalues and eigenfunctions 112

Chapter 3．FREDHOLM INTEGRAL EQUATIONS 115

1．Fredholm Alternative Theorem 115

27．The case when ∫ba∫ba|K(s，t)|2 ds dt<1 115

28．The general case 118

29．Fredholm＇s alternative theorem 125

2．The Schmidt Expansion Theorem and the Mercer Expansion Theorem 127

30．Operator-theoretical notations 127

31．The Schmidt expansion theorem 128

32．Application to Fredholm integral equation of the first kind 131

33．Positive definite kernels．Mercer＇s expansion theorem 132

3．Singular Integral Equations 139

34．Discontinuous kernels 140

35．Examples．Band spectrum 141

Chapter 4．VOLTERRA INTEGRAL EQUATIONS 145

1．Volterra Integral Equations of the Second Kind 145

36．Existence and uniqueness of solutions 145

37．Resolvent kernels 147

38．Application to linear differential equations 149

39．The singular kernel P(s，t)/(s，t)a 151

2．Volterra Integral Equations of the First Kind 153

40．Reduction to integral equations of the second kind 153

41．Abel integral equations 154

Chapter 5．THE GENERAL EXPANSION THEOREM(WEYL-STONE-TITCHMARSH-KODAIRA＇S THEOREM) 159

1．Classification of Singular Boundary Points 160

42．Green＇s formula 160

43．Limit point case and limit circle case 162

44．Definition of m1(λ) and m2(λ) 170

2．The General Expansion Theorem 173

45．Application of the Hilbert-Schmidt expansion theorem 173

46．Helly＇s theorem and Poisson＇s integral formula 177

47．The Weyl-Stone-Titchmarsh-Kodaira theorem 183

48．Density matrix 190

3．Examples 192

49．The Fourier series expansion 192

50．The Fourier integral theorem 194

51．The Hermite function expansion 196

52．The Hankel integral theorem 199

53．The Fourier-Bessel series expansion 202

54．The Laguerre function expansion 205

Chapter 6．NON-LINEAR INTEGRAL EQUATIONS 209

55．Non-linear Volterra integral equations 209

56．Non-linear Fredholm integral equations 210

Appendix．FROM THE THEORY OF FUNCTIONS OF A COMPLEX VARIABLE 213

A theorem on normal family of regular functions(Part 44) 213

Hurwitz＇s theorem(Part 47) 213

The Poisson integral formula(Part 46) 214

BIBLIOGRAPHY 217

INDEX 219